Some authors define the binomial transform with an extra sign, so that it is not self-inverse:
whose inverse is
In this case the former transform is called the inverse binomial transform, and the latter is just binomial transform. This is standard usage for example in On-Line Encyclopedia of Integer Sequences.
ExampleEdit
Both versions of the binomial transform appear in difference tables. Consider the following difference table:
0
1
10
63
324
1485
1
9
53
261
1161
8
44
208
900
36
164
692
128
528
400
Each line is the difference of the previous line. (The n-th number in the m-th line is am,n = 3n−2(2m+1n2 + 2m(1+6m)n + 2m-19m2), and the difference equation am+1,n = am,n+1 - am,n holds.)
The top line read from left to right is {an} = 0, 1, 10, 63, 324, 1485, ... The diagonal with the same starting point 0 is {tn} = 0, 1, 8, 36, 128, 400, ... {tn} is the noninvolutive binomial transform of {an}.
The top line read from right to left is {bn} = 1485, 324, 63, 10, 1, 0, ... The cross-diagonal with the same starting point 1485 is {sn} = 1485, 1161, 900, 692, 528, 400, ... {sn} is the involutive binomial transform of {bn}.
The relationship between the ordinary generating functions is sometimes called the Euler transform. It commonly makes its appearance in one of two different ways. In one form, it is used to accelerate the convergence of an alternating series. That is, one has the identity
which is obtained by substituting x = 1/2 into the last formula above. The terms on the right hand side typically become much smaller, much more rapidly, thus allowing rapid numerical summation.
The Euler transform can be generalized (Borisov B. and Shkodrov V., 2007):
where p = 0, 1, 2,…
The Euler transform is also frequently applied to the Euler hypergeometric integral. Here, the Euler transform takes the form:
The binomial transform, and its variation as the Euler transform, is notable for its connection to the continued fraction representation of a number. Let have the continued fraction representation
The Borel transform will convert the ordinary generating function to the exponential generating function.
Integral representationEdit
When the sequence can be interpolated by a complex analytic function, then the binomial transform of the sequence can be represented by means of a Nörlund–Rice integral on the interpolating function.
GeneralizationsEdit
Prodinger gives a related, modular-like transformation: letting
gives
where U and B are the ordinary generating functions associated with the series and , respectively.
The rising k-binomial transform is sometimes defined as
In the case where the binomial transform is defined as
Let this be equal to the function
If a new forward difference table is made and the first elements from each row of this table are taken to form a new sequence , then the second binomial transform of the original sequence is,
If the same process is repeated k times, then it follows that,